Uniqueness and stability of Ricci flow through singularities

We verify a conjecture of Perelman, which states that there exists a Ricci flow through singularities starting from an arbitrary compact 3-manifold. Our main result is a uniqueness theorem for such flows, , together with an earlier existence theorem of Lott and the second named, implies Perelman's conjecture. We also show that this flow through depends continuously on its initial condition and that it may be as a limit of Ricci flows with surgery. Our results have applications to the study of diffeomorphism groups of three --- in particular to the Generalized Smale Conjecture --- which will in a subsequent paper.